Obama shakes hands with Xi, when the latter was vice-president, in 2012. Xi, now president, meets Obama this week in California. Photograph: Susan Walsh/AP

On Friday the new Chinese president, Xi Jinping, and the United States president, Barack Obama, will meet for two days of talks at Sunnylands, a private estate near Los Angeles. It will be their first meeting since Xi assumed the presidency. The future fortunes of the world are bound up with the two countries finding a new kind of modus vivendi. It will not be easy.

We are living through an extraordinary shift of power from the United States, which has been long dominant, to China, which many now accept will be the dominant power of the future. As has frequently been observed, such shifts are generally the cause of great instability and have often led to conflict.

There have certainly been worrying signs of a deterioration in their relationship. Most important, in this context, has been the so-called US pivot towards east Asia that began in 2010. So far it has been overwhelmingly military in character – involving the deployment of new weapons systems, the strengthening of America's military alliances with Japan, South Korea and the Philippines, and the stationing of marines in northern Australia. This has had the effect of emboldening the Philippines and Vietnam over their claims to various islands in the South China Sea that are contested by China.

The non-military component of the Asian pivot, or "rebalancing" as the Americans now call it, has been the initiative for a new trade bloc – the Trans Pacific Partnership – whose terms are manifestly an attempt to exclude China. With good reason, the Chinese have seen the pivot as an attempt to contain China in its own backyard.

East Asia touches on a crucial question for the future of Sino-American relations. The US has for long been the dominant superpower. Its relations with all other countries, be they friend or foe, have been profoundly unequal. But China's rise increasingly requires the United States to treat it as an equal.

The pivot shows how hard it is for the United States to accept this even in China's own region, where the latter is already the dominant economic player. As has been the case with imperial powers down the ages, the United States will find it extraordinarily difficult to accept a more modest role. But if it refuses to do so, the relationship between the countries will inevitably become increasingly tense.

The history of their bilateral relationship offers, up to a point, significant encouragement. In one of the most dramatic and far-sighted diplomatic initiatives of the past century, the 1972 Nixon-Kissinger rapprochement with China transformed relations. And ever since, notwithstanding the many twists and turns, the relationship between the United States and China has remained remarkably stable. Now, though, there is a new challenge: the relationship must become increasingly equal.

Alas, the response of the Obama administration has not been encouraging. It seems determined to maintain the relationship on its present basis, as if nothing has changed and America remains pre-eminent. This is the recipe for a new cold war. Rather than an attempt to cling to the past, what is required is a major strategic initiative by the two countries that seeks to put their relationship on a new kind of footing. With Obama beginning his second term and Xi commencing what should be a 10-year period in office, this is the opportune moment for them to think big and bold.

The point of departure is their enormously important economic relationship, which has been growing extremely rapidly. In 2000 trade between the two powers was worth $116bn; by 2012 it had reached $536bn. Their economic relationship has the potential to grow hugely in the future and broaden into new fields. China, for example, could provide large amounts of capital that America needs in order to renew its infrastructure. America in turn could lift some of its restrictions on selling hi-tech equipment to China, treating the latter not as a security threat – the rationale for the ban – but as a partner.

The danger is that these possibilities are squandered as relations between the two become mired in growing tensions over cyber attacks, the Korean peninsula, access to the Chinese market, conflicts in the South and East China Seas, human rights, and much else besides. Apart from courage and vision on the part of the two leaders, two other things are required.

First, the Chinese leadership needs to continue to display the same kind of patience and humility that it has demonstrated over the past 30 years. If China became increasingly insistent and inflexible in its demands, reflecting a growing hubris about its own power, the possibilities of the two countries enjoying a relationship of close co-operation would be scuppered.

Second, the United States must adjust to the idea that China is rapidly becoming its equal. It must come to recognise over time that its best interests lie not in resisting or countering China's rise, but in accepting it as a fact and accommodating itself to it. The era when the United States was the dominant global power is steadily coming to an end, and it must find a way of acknowledging this and framing its ambitions and interests accordingly.

Instead of claiming the right to continuing primacy in east Asia, for example, it should seek to share that primacy with China. At present, the United States is a long way from thinking like this: it is even largely in denial that China is about to become the largest economy in the world.

It remains to be seen whether the more optimistic scenario for Sino-American relations can be realised. Much rests on the shoulders of the two leaders: Obama on this score has so far been disappointing; the early signs are that Xi is a highly confident leader who thinks big. The outcome remains, at best, very much in the balance.